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Theorem infiso 8054
Description: Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)
Hypotheses
Ref Expression
infiso.1  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
infiso.2  |-  ( ph  ->  C  C_  A )
infiso.3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )
infiso.4  |-  ( ph  ->  R  Or  A )
Assertion
Ref Expression
infiso  |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
Distinct variable groups:    x, A, y, z    x, B, y, z    x, C, y, z    x, F, y, z    x, R, y, z    x, S, y, z    ph, x, y, z

Proof of Theorem infiso
StepHypRef Expression
1 infiso.1 . . . 4  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
2 isocnv2 6252 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  <-> 
F  Isom  `' R ,  `' S ( A ,  B ) )
31, 2sylib 201 . . 3  |-  ( ph  ->  F  Isom  `' R ,  `' S ( A ,  B ) )
4 infiso.2 . . 3  |-  ( ph  ->  C  C_  A )
5 infiso.4 . . . 4  |-  ( ph  ->  R  Or  A )
6 infiso.3 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )
75, 6infcllem 8034 . . 3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x `' R y  /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  C  y `' R z ) ) )
8 cnvso 5398 . . . 4  |-  ( R  Or  A  <->  `' R  Or  A )
95, 8sylib 201 . . 3  |-  ( ph  ->  `' R  Or  A
)
103, 4, 7, 9supiso 8022 . 2  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  `' S
)  =  ( F `
 sup ( C ,  A ,  `' R ) ) )
11 df-inf 7988 . 2  |- inf ( ( F " C ) ,  B ,  S
)  =  sup (
( F " C
) ,  B ,  `' S )
12 df-inf 7988 . . 3  |- inf ( C ,  A ,  R
)  =  sup ( C ,  A ,  `' R )
1312fveq2i 5895 . 2  |-  ( F `
inf ( C ,  A ,  R )
)  =  ( F `
 sup ( C ,  A ,  `' R ) )
1410, 11, 133eqtr4g 2521 1  |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455   A.wral 2749   E.wrex 2750    C_ wss 3416   class class class wbr 4418    Or wor 4776   `'ccnv 4855   "cima 4859   ` cfv 5605    Isom wiso 5606   supcsup 7985  infcinf 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-po 4777  df-so 4778  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-sup 7987  df-inf 7988
This theorem is referenced by: (None)
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